Drinfeld Orbifold Algebras, with Sarah
Witherspoon.Submitted. Abstract: We
define Drinfeld orbifold algebras as filtered algebras deforming the
skew group algebra (semi-direct product) arising from the action of a
finite group on a polynomial ring. They simultaneously generalize
Weyl algebras, graded (or Drinfeld) Hecke algebras, rational Cherednik
algebras, symplectic reflection algebras, and universal enveloping
algebras of Lie algebras with group actions. We give necessary and
sufficient conditions on defining parameters to obtain Drinfeld
orbifold algebras in two general formats, both algebraic and
homological. We explain the connection between Hochschild
cohomology and a Poincare-Birkhoff-Witt property explicitly (using
Gerstenhaber brackets). We also classify those deformations
of skew group algebras which arise as Drinfeld orbifold algebras and
give applications for abelian groups.

Quantum Drinfeld Hecke Algebras, with Viktor Levandovskky. Submitted. Abstract: We
consider finite groups acting on quantum (or skew) polynomial
rings. Deformations of the semidirect product of the quantum
polynomial ring with the acting group extend symplectic reflection
algebras and graded Hecke algebras to the quantum setting over a field
of arbitrary characteristic. We give necessary and sufficient
conditions for such algebras to satisfy a Poincare-Birkhoff-Witt
property using the theory of noncommutative Groebner bases. We include
applications to the case of abelian groups and the case of groups
acting on coordinate rings of quantum planes. In addition, we classify
graded automorphisms of the coordinate ring of quantum 3-space.
In characteristic zero, Hochschild cohomology gives an elegant
description of the PBW conditions.

Group Actions on Algebras and the Graded Lie Structure of Hochschild Cohomology, with Sarah
Witherspoon. Journal of Algebra, in press. Abstract:
Hochschild cohomology governs deformations of algebras, and its
graded Lie structure plays a vital role. We study this structure for
the Hochschild cohomology of the skew group algebra formed by a finite
group acting on an algebra by automorphisms.We examine the Gerstenhaber
bracket with a view toward deformations and developing
bracket formulas. We then focus on the linear group actions and
polynomial algebras that arise in orbifold theory and representation
theory; deformations in this context include graded Hecke algebras and
symplectic reflection algebras. We give some general results
describing when brackets are zero for polynomial skew group algebras,
which allow us in particular to find noncommutative Poisson structures.
For abelian groups, we express the bracket using inner products of
group characters. Lastly, we interpret results for graded Hecke
algebras.

Finite groups actings
linearly: Hochschild cohomology and the cup product, with Sarah
Witherspoon. Advances in Mathematics, 226 (4), 2011, 2884--2910. Abstract:
When a finite group acts linearly on a complex vector space, the
natural semi-direct product of the group and the polynomial ring over
the space forms a skew group algebra. This algebra plays the role of
the coordinate ring of the resulting orbifold and serves as a
substitute for the ring of invariant polynomials from the viewpoint of
geometry and physics. Its Hochschild cohomology predicts various
Hecke algebras and deformations of the orbifold. In this article, we
investigate the ring structure of the Hochschild cohomology of the skew
group algebra. We show that the cup product coincides with a natural
smash product, transferring the cohomology of a group action into a
group action on cohomology. We express the algebraic structure of
Hochschild cohomology in terms of a partial order on the group (modulo
the kernel of the action). This partial order arises after
assigning to each group element the codimension of its fixed point
space. We describe the algebraic structure for Coxeter groups,
where this partial order is given by the reflection length function; a
similar combinatorial description holds for an infinite family of
complex reflection groups.

Quantum differentiation
and chain maps of bimodule complexes, with Sarah Witherspoon.
Algebra and Number Theory. 5-3 (2011), 339-360. Abstract:
We
consider a finite group acting on a vector space and the corresponding
skew group algebra generated by the group and the symmetric algebra of
the space. This skew group algebra illuminates the resulting orbifold
and serves as a replacement for the ring of invariant polynomials,
especially in the eyes of cohomology. One analyzes the Hochschild
cohomology of the skew group algebra using isomorphisms which convert
between resolutions. We present an explicit chain map from the bar
resolution to the Koszul resolution of the symmetric algebra which
induces various isomorphisms on Hochschild homology and cohomology,
some of which have appeared in the literature before. This approach
unifies previous results on homology and cohomology of both the
symmetric algebra and skew group algebra. We determine induced
combinatorial cochain maps which invoke quantum differentiation
(expressed by Demazure-BBG operators).

Jacobians of Reflection Groups. With Julia Hartmann. Transactions of the American
Mathematical Society, 360 (2008), no.1,
123--133. Abstract:
Steinberg showed that when a finite reflection group acts on a real
or complex vector space of finite dimension, the Jacobian determinant of
a set of basic invariants factors into linear forms which define the reflecting
hyperplanes. This result generalizes verbatim to fields whose characteristic
is prime to the order of the group. Our main theorem gives a generalization
of Steinberg's result for arbitrary fields, using a ramification formula
of Benson and Crawley-Boevey. As an intermediate result, we show that every
finite group which fixes a hyperplane pointwise has a polynomial ring of
invariants.

Hochschild cohomology
and graded Hecke algebras, with Sarah Witherspoon. Transactions
of the American
Mathematical Society 360
(2008), no. 8, 3975--4005. Abstract:
We develop and collect techniques for determining Hochschildcohomology
of skew group algebras S(V) # G and apply our results to graded Hecke
algebras. We discuss the explicit computation of certain types of
invariants under centralizer subgroups, focusing on the infinite family
of complex reflection groups G(r,p,n) to illustrate our ideas.
Resulting formulas for Hochschild two-cocycles give information about
deformations of S(V) # G and, in particular, about graded Hecke
algebras. We expand the definition of a graded Hecke algebra to allow a
nonfaithful action of G on V, and we show that there exist
nontrivial graded Hecke algebras for G(r,1,n), in contrast to the
case of the natural reflection representation. We prove that one of
these graded Hecke algebras is equivalent to an algebra that has
appeared before in a different form.

Reflection
groups and differential forms, with Julia
Hartmann. Mathematical Research Letters 14
(2007), no.6, 955--971. Abstract:
Steinberg
showed that when a finite reflection group acts on a real or complex
vector space of finite dimension, the Jacobian determinant of a set of
basic invariants factors into linear forms which define the reflecting
hyperplanes. This result generalizes verbatim to fields whose
characteristic is prime to the order of the group. Our main theorem
gives a generalization of Steinberg's result for groups with polynomial
ring of invariants over arbitrary fields using a ramification formula
of Benson and Crawley-Boevey.

Generalized Exponents and Forms. dviJournal of Algebraic Combinatorics, 22
(2005),no. 1, 115--132.Abstract:
We consider generalized exponents of a finite reflection group acting
on a real or complex vector space V. These integers are the degrees in
which an irreducible representation of the group occurs in the coinvariant
algebra. A basis for each isotypic component arises in a natural way from
a basis of invariant generalized forms. We investigate twisted reflection
representations (V tensor a linear character) using the theory of semi-invariant
differential forms. Springer's theory of regular numbers gives a formula
when the group is generated by dim V reflections. Although our arguments
are case-free, we also include explicit data and give a method (using differential
operators) for computing semi-invariants and basic derivations. The data
give bases for certain isotypic components of the coinvariant algebra.

Graded Hecke Algebras for Complex Reflection Groups, with Arun
Ram. Commentairi M. Helvetici, 78, 308-334, 2003. dvi,ps. Abstract:
The graded Hecke algebra for a finite Weyl group is intimately related
to the geometry of the Springer correspondence. A construction of
Drinfeld produces an analogue of a graded Hecke algebra for any finite
subgroup of GL(V). This paper classifies all the algebras obtained
by applying Drinfeld's construction to complex reflection groups.
By giving explicit (though nontrivial) isomorphisms, we show that the graded
Hecke algebras for finite real reflection groups constructed by Lusztig
are all isomorphic to algebras obtained by Drinfeld's construction.
The classification shows that there exist algebras obtained from Drinfeld's
construction which are not graded Hecke algebras as defined by Lusztig
for real as well as complex reflection groups.

The Sign representation for Shephard Groups, with Peter Orlik and
Victor
Reiner.Mathematische Annalen, 322, p. 477-492, 2002.
ps,
dviAbstract:
Shephard groups are unitary reflection groups arising as the symmetries
of regular complex polytopes. For a Shephard group, we identify the
representation carried by the principal ideal in the coinvariant algebra
generated by the image of the product of all linear forms defining reflecting
hyperplanes. This representation turns out to have many equivalent
guises making it analogous to the sign representation of a finite Coxeter
group. One of these guises is (up to a twist) the cohomology of the
Milnor fiber for the isolated singularity at 0 in the hypersurface defined
by any homogeneous invariant of minimal degree.

Logarithmic
forms and anti-invariant forms of reflection groups, with Hiroaki
Terao. In Advanced Studies in Pure Math., 27,
Arrangements, Tokyo, 1998, 2000, pdf,
psAbstract:
Let W be a finite group generated by unitary reflections and A be the
set of reflecting hyperplanes. We will give a characterization of
the logarithmic differential forms with poles along A in terms of anti-invariant
differential forms. If W is a Coxeter group defined over the real
numbers, then the characterization provides a new method to find a basis
for the module of logarithmic differential forms out of basic invariants.

Let G be a finite group of complex n x n unitary matrices generated
by reflections acting on C^n. Let R be the ring of invariant polynomials,
and let \chi be a multiplicative character of G. Let \Omega^\chi
be the R-module of \chi-invariant differential forms. We define a
multiplication in \Omega^\chi and show that under this multiplication \Omega^\chi
has an exterior algebra structure. We also show how to extend
the results to vector fields, and exhibit a relationship between \chi-invariant
forms and logarithmic forms.